The motion that measures faster than light

Photograph a distant radio source, wait a year, photograph it again, and measure how far a bright knot has moved. Divide by the distance to the source to get an angle, by a year to get a rate, and the result is a transverse speed — a proper motion turned into a velocity in the ordinary way.

For several dozen sources that speed comes out greater than one. In the best-measured cases it is around six.

Where the excess comes from #

Nothing about the source’s motion is at fault. What is at fault is dividing by the wrong interval.

Suppose the knot moves at speed $\beta$ at an angle $\theta$ to the line of sight, and consider two flashes separated by a time $t$ in the observer’s frame. In that time the knot moves $\beta t \sin\theta$ across the sky and $\beta t\cos\theta$ towards the observer.

The second flash therefore has a shorter journey, by $\beta t\cos\theta$, so it arrives early by that amount. The interval between arrivals is

$$t_{\text{obs}} = t\,(1 - \beta\cos\theta)$$

and dividing the transverse distance by it gives

$$\beta_{\text{app}} = \frac{\beta\sin\theta}{1 - \beta\cos\theta}.$$

The denominator can be small. That is the whole of it, and no relativity has been used: the same arithmetic would apply to a slow object if light travelled slowly enough to notice.

Two features of the expression are worth extracting, and the figure locates both on the drawn curves rather than asserting them.

The maximum over angle is $\gamma\beta$, reached where $\cos\theta = \beta$. So the largest illusion available to a source depends on its speed alone, and getting an apparent ten requires $\gamma$ of at least ten.

Below $\beta = 1/\sqrt2$ there is no angle that works. The effect has a threshold, and a source showing it is a source travelling above seven-tenths of the speed of light.

It is worth stating the size of the correction in the everyday case, because it is the same formula. A car approaching at thirty metres a second and passing at ten degrees to the line of sight has its apparent transverse speed altered by a part in ten million, which is why nobody has ever noticed. The formula does not have a regime where it switches on; it has a factor that is unmeasurable until the source is fast.

What an observation of it buys #

The illusion is usually presented as a warning. It is better read as an instrument, because it constrains two quantities at once and nothing else does.

Observing an apparent transverse speed $\beta_{\text{app}}$ requires

$$\beta \ge \frac{\beta_{\text{app}}}{\sqrt{1 + \beta_{\text{app}}^2}},$$

which for an apparent $7$ is a true speed of at least $0.990$, and it also bounds the angle: a source that fast, seen at an angle much larger than $\arccos\beta$, would show a smaller apparent speed. An observed $7.02$ therefore puts a floor under the speed and a ceiling of about sixteen degrees on the orientation.

That is how the Lorentz factors of active galactic nuclei are estimated, and the numbers that come out — tens, occasionally more — are otherwise unobtainable. A source at a hundred megaparsecs cannot be timed any other way.

There is a corroborating measurement, from a different effect with the same geometry in it. A source moving towards the observer is brightened by the fourth power of its Doppler factor, so a jet seen at a small angle is enormously brighter on the approaching side than on the receding one — which is why so many are seen as one-sided. Combining the brightness ratio with the apparent speed over-determines the pair of unknowns, and the two methods agree.

There is a way of putting the whole effect that removes the mystery entirely, and it is worth having. Suppose a runner sets off towards a stationary observer and shouts once a second by his own watch. The shouts arrive faster than once a second, because each one starts closer. Nobody finds that puzzling and nobody concludes the runner is talking quickly. Now let the runner move mostly towards the observer and slightly across, and time his sideways progress by the arrival of the shouts: the sideways distance is real and the interval is compressed, so the rate is too large. That is the entire content, with sound instead of light and a runner instead of a jet, and the only reason the astronomical version is startling is that the compression factor there can be a hundred.

The same illusion, closer to home #

The effect is not confined to distant galaxies, and one instance was watched in real time.

An object inside this galaxy, ejecting material at nearly the speed of light and doing so on a timescale of days rather than years, was seen in 1994 to throw out two blobs in opposite directions. One of them appeared to move at $1.25$ times the speed of light and the other at $0.65$.

Because the two are back to back, the geometry is over-determined: the pair of apparent speeds gives both the true speed and the angle without any further assumption, and the answer was $0.92$ at $70$ degrees. That is the effect used as a measurement rather than as a curiosity, and it worked because there were two blobs rather than one.

The same arithmetic explains the asymmetry that makes one-sided jets so common. The receding side’s apparent speed is $\beta\sin\theta/(1 + \beta\cos\theta)$ — the sign in the denominator flips — which is always less than the true speed and, for a small angle, very much less. So a symmetric pair of jets looks like a fast one and a slow one, and if the receding one is also faint enough to be lost, like one jet.

The history, and the argument that was settled by it #

The effect was predicted before it was seen, which is unusual for something that looks like a mistake.

Nothing about the superluminal illusion moves an event from one causal class to another, which is the formal statement that no signal is outrunning anything. The intervals between the emissions are what they were; only the arrival times have been compressed, because the source moved toward the observer between one emission and the next. Causality is a statement about intervals and the illusion is a statement about arrivals.

Martin Rees published the geometry in 1966, as a way of accounting for radio sources that appeared to vary faster than their light-crossing times allowed. Very-long-baseline interferometry began resolving quasar structure a few years later and the motions were seen, at apparent speeds of several times light’s, in 1971.

The reception was mixed, and the reason is worth recording. A number of alternatives were proposed — that the sources were much nearer than their redshifts implied, that new physics was involved, that the moving features were illusions of a different kind — and each of them was a way of avoiding a conclusion the observers were not yet sure of. The geometrical explanation had the disadvantage of requiring bulk motion at more than ninety per cent of the speed of light, which in 1971 nobody had any independent reason to expect — a Lorentz factor of ten being, at the time, a laboratory number rather than an astronomical one.

What settled it was the accumulation of the corroborating measurement described above. Jets that show large apparent speeds are also the ones with one-sided structure, with strong variability, and with the high brightness temperatures that beaming implies — and every one of those is what a source pointed nearly at the observer should look like. A single strange observation became four consistent ones, all requiring the same two numbers.

The lesson is a general one about inference from images of distant things. What is measured is the pattern of arrival times of light, and turning that into a description of what happened requires a model of the geometry. Where the source is fast and the geometry is unknown, the same data support very different pictures, and the way out is always another observable that depends on the geometry differently.

What is really being timed #

It is worth being clear about which quantity has been mis-measured, because “the light took less time to arrive” can sound like an excuse.

The effect is not the relativity of simultaneity, and it is worth saying so because the two are routinely confused. What is really being timed is a difference of light-travel times, and it would exist in a theory with no relativity in it at all — a nineteenth-century physicist with a finite speed of light would have predicted it. What relativity adds is a ceiling on the source’s speed, and therefore a ceiling on how large the apparent speed can be.

Nothing here is about simultaneity, or about frames, or about anything that changed in 1905. The observer has one frame, in which the knot travels at less than light speed, in which the two flashes are separated by a definite time, and in which the light of each takes a definite time to arrive. What the observer records is the difference of the arrival times, which is not the difference of the emission times, and dividing by it gives a rate that is not a speed.

Read that way the effect is a cousin of the way a fast object photographs as rotated rather than contracted: both come from light from different parts of an event reaching the camera at different times, and both would occur in a universe with a finite light speed and no relativity whatever.

Relativity enters only in setting the ceiling. Without it there would be no reason for $\beta$ to be less than one, and the apparent speed would carry no information at all.

Two blobs, and the arithmetic that closes #

The over-determined case is worth working through, because it turns the illusion into a complete measurement with nothing assumed.

Take a source that ejects material both ways at once, at the same speed. The approaching side shows

$$\beta_+ = \frac{\beta\sin\theta}{1 - \beta\cos\theta}, \qquad \beta_- = \frac{\beta\sin\theta}{1 + \beta\cos\theta}.$$

Two measurements, two unknowns. Combining them gives

$$\beta\cos\theta = \frac{\beta_+ - \beta_-}{\beta_+ + \beta_-},$$

and a second combination gives $\beta\sin\theta$, so both the speed and the angle follow without any further assumption. That is what happened with the galactic source: apparent speeds of $1.25$ and $0.65$ gave a true speed of $0.92$ and an angle of $70$ degrees.

Notice what the angle turned out to be. The illusion is usually described as requiring a jet pointed nearly at the observer, and this one is pointed nowhere near — it produced an apparent superluminal motion at seventy degrees because $1.25$ is only just above one and the threshold is generous once the speed is high. The requirement is on $\gamma\beta$ exceeding the apparent speed, not on the angle being small.

What the phrase is worth being careful about #

“Apparent superluminal motion” is a phrase that has caused more confusion than the effect, and the confusion is worth naming because a version of it recurs whenever something distant is measured.

The measurement is not wrong. Two positions were recorded correctly, two arrival times were recorded correctly, and their ratio was computed correctly. What is wrong is a description attached to the ratio — that it is a speed — and that description is an inference from the data rather than part of it.

The general shape is: an instrument reports a number; the number is turned into a physical quantity by a chain of assumptions; and one of the assumptions is that the interval between two arrivals is the interval between two departures. That assumption is exact only for a source at a fixed distance. It fails for anything approaching, by a factor of $1 - \beta\cos\theta$, and for a fast source that factor is not a correction but the dominant term.

The same care is needed for every quantity read off a distant object’s light curve. A variability timescale bounds a source’s size only after the same factor has been divided out, and doing so is what turns an apparently impossible brightness temperature into an ordinary one. The correction is a single number, it applies to a whole family of measurements at once, and it is why the Doppler factor rather than the velocity is what such observations are quoted in.

The version in which nothing moves at all #

There is a second family of apparently superluminal motions, and keeping it separate from this one is worth the trouble, because the two are usually run together and they are not the same illusion.

In everything above, something really is moving, and moving fast — the floor under the true speed is the whole value of the observation. In the other family nothing moves at any speed whatever.

Sweep a laser pointer across the face of the Moon. The angular rate that a wrist can produce is of order a radian per second, and the Moon is four hundred million metres away, so the illuminated spot crosses its surface at more than the speed of light. Nothing has been transported: each photon travelled outward at $c$, and the spot is a name for wherever the current batch happens to land. Two people standing on the Moon in the spot’s path cannot use it to send each other anything, because what arrives at the second was emitted from the pointer and not from the first. The intersection of a pair of closing scissor blades does the same trick, and so does the crest of a wave whose phase velocity carries no signal.

The astronomical instance of that family is a light echo, and it is spectacular. A star that flashes once illuminates the dust around it, and the dust scatters some of that light towards the observer. The grains that can be seen at any given moment are the ones for which the path from star to grain to observer takes exactly the elapsed time — a paraboloid with the star at its focus and the line of sight as its axis — and as time passes that paraboloid sweeps outward through the dust.

What is seen is a bright ring or shell expanding across the sky, and for dust lying in front of the star it expands faster than light. The star V838 Monocerotis flared in 2002 and was photographed for years afterwards apparently blowing out an enormous shell; the shell was never there. The dust was already in place and stationary, and what moved through it was the illumination.

The two families are told apart by exactly the reasoning of this essay, and by one more observable. A real jet’s knots are self-luminous and keep their spectra; a light echo’s brightening is scattered light, so it carries the flash’s spectrum wherever it appears, arriving unchanged at places that could not have communicated. That difference is decisive, and it turns the echo into an instrument of its own: because the geometry of the paraboloid is fixed by the elapsed time and by nothing else, measuring how fast the ring grows against how long ago the flash occurred gives the distance to the star, with no ladder of standard candles anywhere in the argument.

So the same headline covers two quite different situations, and the discipline is the one the previous section named. Ask what the moving thing is. If it is matter, an apparent superluminal speed is a measurement of how fast that matter is going. If it is a place where a condition happens to be satisfied, it is a measurement of nothing but geometry, and there is no floor under anything.

Where the model runs out #

The blob is treated as a single moving object and probably is not. What is seen brighten and move may be a pattern rather than material — a shock travelling through a slower flow, whose pattern speed is not the flow speed. Then the inferred Lorentz factor belongs to the pattern, and the underlying flow may be faster or slower — a pattern speed being under no obligation to be a speed of anything.

Every source showing an apparent superluminal motion is also being brightened enormously, and the two are the same geometry: a source moving nearly along the line of sight is both apparently fast and apparently bright. That is why the effect is seen so often in quasar jets despite requiring a narrow alignment — the alignment that produces it also selects those objects into the sample.

Selection is severe and works in the same direction. A source has to be beamed towards the observer to be bright enough to notice, and beaming towards the observer is exactly the geometry that produces the illusion. The population that is seen to move superluminally is therefore not a fair sample of jets, and correcting for that is a substantial part of the work.

Acceleration is ignored. The derivation is for constant velocity, and jets accelerate: measured apparent speeds increase along the length of some of them. The instantaneous relation still holds, but the inferred angle and speed are then functions of position rather than properties of the source.

And the distance has to come from somewhere else. The measurement is an angle per unit time, and turning it into a speed requires the distance to the source — which for a cosmological object comes from a redshift and a cosmological model. An error in the model is an error in every apparent speed derived with it, in proportion — which is why the effect is quoted as a Lorentz factor rather than as a velocity wherever the two can be separated, and why what a redshift means has to be settled first.

The ladder from here #

Later rungs on this anchor: the Doppler factor as the single quantity that both the brightness and the timing depend on, and how the two measurements are combined; jet acceleration and the velocity gradients that measured apparent speeds imply; the counter-jet ratio as an independent constraint on the same geometry; and the pattern-versus-flow ambiguity, which is where the largest systematic uncertainty in the whole subject sits.

The neighbouring ladders are speeds that refuse to add, which is where the ceiling on the true speed comes from, the sky that crowds into a cone, which is what the same geometry does to directions, and now is a choice of slicing, which is the thing this effect is repeatedly and wrongly said to be about.